Integrand size = 28, antiderivative size = 384 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 (e x)^{3/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{3/2} \sqrt {c+d x^2}}{5 d^2 e}-\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{5 c d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 c^{3/4} d^{11/4} \sqrt {c+d x^2}}-\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{10 c^{3/4} d^{11/4} \sqrt {c+d x^2}} \]
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Time = 0.23 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {474, 470, 335, 311, 226, 1210} \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {\sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{10 c^{3/4} d^{11/4} \sqrt {c+d x^2}}+\frac {\sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 c^{3/4} d^{11/4} \sqrt {c+d x^2}}-\frac {\sqrt {e x} \sqrt {c+d x^2} \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right )}{5 c d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{3/2} \sqrt {c+d x^2}}{5 d^2 e} \]
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Rule 226
Rule 311
Rule 335
Rule 470
Rule 474
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 (e x)^{3/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {e x} \left (\frac {1}{2} \left (-2 a^2 d^2+3 (b c-a d)^2\right )-b^2 c d x^2\right )}{\sqrt {c+d x^2}} \, dx}{c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{3/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{3/2} \sqrt {c+d x^2}}{5 d^2 e}-\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{10 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{3/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{3/2} \sqrt {c+d x^2}}{5 d^2 e}-\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 c d^2 e} \\ & = \frac {(b c-a d)^2 (e x)^{3/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{3/2} \sqrt {c+d x^2}}{5 d^2 e}-\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 \sqrt {c} d^{5/2}}+\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 \sqrt {c} d^{5/2}} \\ & = \frac {(b c-a d)^2 (e x)^{3/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{3/2} \sqrt {c+d x^2}}{5 d^2 e}-\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{5 c d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 c^{3/4} d^{11/4} \sqrt {c+d x^2}}-\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 c^{3/4} d^{11/4} \sqrt {c+d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.31 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {e x} \left (x \left (-10 a b c d+5 a^2 d^2+b^2 c \left (7 c+2 d x^2\right )\right )+\left (-21 b^2 c^2+30 a b c d-5 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{5 c d^2 \sqrt {c+d x^2}} \]
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Time = 4.08 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.83
method | result | size |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {e \,x^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{2} c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} x \sqrt {d e \,x^{3}+c e x}}{5 d^{2}}+\frac {\left (\frac {b \left (2 a d -b c \right ) e}{d^{2}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) e}{2 d^{2} c}-\frac {3 b^{2} c e}{5 d^{2}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(319\) |
risch | \(\frac {2 b^{2} x^{2} \sqrt {d \,x^{2}+c}\, e}{5 d^{2} \sqrt {e x}}+\frac {\left (\frac {\left (10 a b d -8 b^{2} c \right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}+\left (5 a^{2} d^{2}-10 a b c d +5 b^{2} c^{2}\right ) \left (\frac {x^{2}}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}-\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e \sqrt {e x \left (d \,x^{2}+c \right )}}{5 d^{2} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(442\) |
default | \(-\frac {\sqrt {e x}\, \left (10 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-60 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +42 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-5 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}+30 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d -21 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-4 b^{2} c \,d^{2} x^{4}-10 a^{2} d^{3} x^{2}+20 a b c \,d^{2} x^{2}-14 b^{2} c^{2} d \,x^{2}\right )}{10 \sqrt {d \,x^{2}+c}\, d^{3} x c}\) | \(597\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left (21 \, b^{2} c^{3} - 30 \, a b c^{2} d + 5 \, a^{2} c d^{2} + {\left (21 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (2 \, b^{2} c d^{2} x^{3} + {\left (7 \, b^{2} c^{2} d - 10 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{5 \, {\left (c d^{4} x^{2} + c^{2} d^{3}\right )}} \]
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\[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e x} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {e x}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {e x}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e\,x}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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